The GA for the flight can add you until the flight is closed. "Closed" does not mean the door, but rather that the paperwork is transmitted to the dispatcher to calculate weight & balance.

If there is a delay and the GA is willing to add you while the aircraft is sitting at the gate and the flight is not closed, the GA can certainly do it.

But, it is completely the agent's discretion. It is going to be low on the agent's list of priorities as delaying the push because the GA has boarded a passenger and loused up the passenger count requiring both the Captain and dispatch to recalculate weight & balance is not the kind of thing which endears GA's to their employer.